Fuzzy Interpretation of the Discharge Coefficient for the Orifice Plate of Automated Flow Rate Measuring Systems

Figures & data

Figure 1. Dependence of the discharge coefficient on the number of Re

Figure 2. Functions that depend on $\beta$, the components of equation (3)(3) $C_d=\overrightarrow{f(\beta )}\cdot \overrightarrow{g(Re)},$(3)

Figure 3. Functions dependent on Re in equation (3)(3) $C_d=\overrightarrow{f(\beta )}\cdot \overrightarrow{g(Re)},$(3)

Figure 4. Fuzzy sets for the variable «Orifice plate diameter ratio»

Figure 5. Fuzzy sets for the variable «Reynolds Number»

Figure 6. The structure of the fuzzy system

Table

R1:	IF $(\beta =A_1)$	THEN $f_1(\beta )=0,596+0,005\beta$ AND $f_3(\beta )=13,38\beta ;$
R2:	IF $(\beta =A_2)$	THEN $f_1(\beta )=0,597+0,023(\beta -0,2)$ AND $f_2(\beta )=0,071+5,643(\beta -0,2)$ AND $f_3(\beta )=2,676+8,023(\beta -0,2)$ AND $f_4(\beta )=-1,48(\beta -0,2)$ AND $f_5(\beta )=0,02+3,31(\beta -0,2);$
R3:	IF $(\beta =A_3)$	THEN $f_1(\beta )=0,604+0,016(\beta -0,5)$ AND $f_2(\beta )=1,764+22,104(\beta -0,5)$ AND $f_3(\beta )=5,083+6,673(\beta -0,5)$ AND $f_4(\beta )=-0,444-15,288(\beta -0,5)$ AND $f_5(\beta )=1,013+19,112(\beta -0,2);$
R4:	IF $(Re=B_1)$	THEN $g_2(Re)=0,015-1,136\cdot {10}^{-6}(Re-100)$ AND $g_3(Re)=0,04-3,88\cdot {10}^{-6}(Re-100)$ AND $g_4(Re)=0,025-2,46\cdot {10}^{-6}(Re-100)$ AND $g_5(Re)=0,333-3,34\cdot {10}^{-5}(Re-100);$
R5:	IF $(Re=B_2)$	THEN $g_2(Re)=3,751\cdot {10}^{-3}-0,526\cdot {10}^{-7}(Re-{10}^4)$ AND $g_3(Re)=1,585\cdot {10}^{-3}-4,252\cdot {10}^{-8}(Re-{10}^4)$ AND $g_4(Re)=6,31\cdot {10}^{-4}-1,845\cdot {10}^{-8}(Re-{10}^4)$ AND $g_5(Re)=2,101\cdot {10}^{-3}-0,737\cdot {10}^{-7}(Re-{10}^4);$
R6:	IF $(Re=B_3)$	THEN $g_2(Re)=2,698\cdot {10}^{-3}-0,117\cdot {10}^{-7}(Re-3\cdot {10}^4)$ AND $g_3(Re)=7,345\cdot {10}^{-4}-0,5975\cdot {10}^{-8}(Re-3\cdot {10}^4)$ AND $g_4(Re)=2,62\cdot {10}^{-4}-0,231\cdot {10}^{-8}(Re-3\cdot {10}^4)$ AND $g_5(Re)=0,6274\cdot {10}^{-3}-0,066\cdot {10}^{-7}(Re-3\cdot {10}^4).$

Figure 7. Functions depending on the diameter ratio included in equation (3)(3) $C_d=\overrightarrow{f(\beta )}\cdot \overrightarrow{g(Re)},$(3) after the fuzzy rule base

Figure 8. Functions dependent on the number Re, included in equation (3)(3) $C_d=\overrightarrow{f(\beta )}\cdot \overrightarrow{g(Re)},$(3) after the fuzzy rule base

Figure 9. The relative error of determination of fuzzy discharge coefficient with various values of $\beta$

Figure 10. Additional fuzzy set for the variable «Reynolds Number»

Figure 11. Functions dependent on Reynolds number in the equation (3)(3) $C_d=\overrightarrow{f(\beta )}\cdot \overrightarrow{g(Re)},$(3) after introduction of additional fuzzy set

Figure 12. Relative error of determination of fuzzy discharge coefficient after introduction of additional fuzzy set

Table

R4:	IF $(Re=B_1)$	THEN $g_2(Re)=0,005686-0,258\cdot {10}^{-6}(Re-2500)$ AND $g_3(Re)=0,004183-0,346\cdot {10}^{-6}(Re-2500)$ AND $g_4(Re)=0,0019-1,71\cdot {10}^{-7}(Re-2500)$ AND $g_5(Re)=0,00965-1,007\cdot {10}^{-6}(Re-2500);$
R7:	IF $(Re=B_0)$	THEN $g_2(Re)=0,015-3,88\cdot {10}^{-6}(Re-100)$ AND $g_3(Re)=0,04-14,92\cdot {10}^{-6}(Re-100)$ AND $g_4(Re)=0,025-9,62\cdot {10}^{-6}(Re-100)$ AND $g_5(Re)=0,333-1,347\cdot {10}^{-4}(Re-100).$

Figure 13. Comparison of different models of the discharge coefficient

Figure 14. Comparison of relative errors in the discharge coefficient of the present model with two previous models